Kamran Majid

Kamran Majid

The classical limit of quantum mechanics, formally investigated through frameworks like strict deformation quantization, remains a profound area of inquiry in the philosophy of physics. This paper explores a computational approach employing a neural network to emulate the emergence of classical behavior from the quantum harmonic oscillator as Planck's constant $\hbar$ approaches zero. We develop and train a neural network architecture to learn the mapping from initial expectation values and $\hbar$ to the time evolution of the expectation value of position. By analyzing the network's predictions across different regimes of hbar, we aim to provide computational insights into the nature of the quantum-classical transition. This work demonstrates the potential of machine learning as a complementary tool for exploring foundational questions in quantum mechanics and its classical limit.

Kamran Majid

The emergence of classical behavior from quantum mechanics as Planck's constant $\hbar$ approaches zero remains a fundamental challenge in physics [1-3]. This paper introduces a novel approach employing deep neural networks to directly learn the dynamical mapping from initial quantum state parameters (for Gaussian wave packets of the one-dimensional harmonic oscillator) and $\hbar$ to the parameters of the time-evolved Wigner function in phase space [4-6]. A comprehensive dataset of analytically derived time-evolved Wigner functions was generated, and a deep feedforward neural network with an enhanced architecture was successfully trained for this prediction task, achieving a final training loss of ~ 0.0390. The network demonstrates a significant and previously unrealized ability to accurately capture the underlying mapping of the Wigner function dynamics. This allows for a direct emulation of the quantum-classical transition by predicting the evolution of phase-space distributions as $\hbar$ is systematically varied. The implications of these findings for providing a new computational lens on the emergence of classicality are discussed, highlighting the potential of this direct phase-space learning approach for studying fundamental aspects of quantum mechanics. This work presents a significant advancement beyond previous efforts that focused on learning observable mappings [7], offering a direct route via the phase-space representation.